Matrix algebra

Let $M=\begin{bmatrix}0&-\alpha\\\alpha&0\end{bmatrix}$, where $\alpha$ is a non-zero real number an $N=\displaystyle\sum_{k=1}^{49}M^{2k}$. If $\left(I-M^2\right)N=-2I$, then the positive integral value of $\alpha$ is ______.

Let $A=\begin{pmatrix}2&-1&-1\\1&0&-1\\1&-1&0\end{pmatrix}$ and $B=A-I$. If $\omega=\dfrac{\sqrt3i-1}2$, then the number of elements in the set $\left\{n\in\{1,2,\dots,100\}:A^n+(\omega B)^n=A+B\right\}$ is equal to ______.

Let $A=\begin{bmatrix}1&a&a\\0&1&b\\0&0&1\end{bmatrix}$, $a,b\in\mathbb R$. If for some $n\in\mathbb N$, $A^n=\begin{bmatrix}1&48&2160\\0&1&96\\0&0&1\end{bmatrix}$ then $n+a+b$ is equal to ______.

Let $A=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $B=\begin{bmatrix}9^2&-10^2&11^2\\12^2&13^2&-14^2\\-15^2&16^2&17^2\end{bmatrix}$, then the value of $A'BA$ is:

Let $A=\begin{pmatrix}1&2\\-2&-5\end{pmatrix}$. Let $\alpha,\beta\in\mathbf R$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha+\beta$ is equal to

Let $A=\begin{pmatrix}4&-2\\\alpha&\beta\end{pmatrix}$.

If $A^2+\gamma A+18I=O$, then $\det(A)$ is equal to ______.

Let $A=\begin{bmatrix}1&-1\\2&\alpha\end{bmatrix}$ and $B=\begin{bmatrix}\beta&1\\1&0\end{bmatrix}$, $\alpha,\beta\in\mathbf R$. Let $\alpha_1$ be the value of $\alpha$ which satisfies $(A+B)^2=A^2+\begin{bmatrix}2&2\\2&2\end{bmatrix}$ and $\alpha_2$ be the value of $\alpha$ which satisfies $(A+B)^2=B^2$. Then $|\alpha_1-\alpha_2|$ is equal to ______.

The number of matrices of order 3 × 3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is _______.

Which of the following matrices can NOT be obtained from the matrix $\begin{bmatrix}-1&2\\1&-1\end{bmatrix}$ by a single elementary row operation ?

Let $X=\begin{bmatrix}1\\1\\1\end{bmatrix}$ and $A=\begin{bmatrix}-1&2&3\\0&1&6\\0&0&-1\end{bmatrix}$. For k ∈ ℕ, if $X'A^kX=33$, then k is equal to _______.

For any $3\times3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let
$$E = \begin{bmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 8 & 13 & 18\end{bmatrix},\quad P = \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix}\quad\text{and}\quad F = \begin{bmatrix}1 & 3 & 2\\ 8 & 18 & 13\\ 2 & 4 & 3\end{bmatrix}$$
If $Q$ is a nonsingular matrix of order $3\times3$, then which of the following statements is (are) TRUE?

For any $3\times3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $I$ be the $3\times3$ identity matrix. Let $E$ and $F$ be two $3\times3$ matrices such that $(I - EF)$ is invertible. If $G = (I - EF)^{-1}$, then which of the following statements is (are) TRUE?

Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of M$^T$M is seven, is _______.

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2\times2$ matrix such that the trace of $A$ is 3 and the trace of $A^3$ is $-18$, then the value of the determinant of $A$ is ______

Let

$$M = \begin{bmatrix}\sin^4\theta & -1-\sin^2\theta\\ 1+\cos^2\theta & \cos^4\theta\end{bmatrix} = \alpha I + \beta M^{-1},$$

where $\alpha = \alpha(\theta)$ and $\beta = \beta(\theta)$ are real numbers, and $I$ is the $2\times2$ identity matrix. If

$\alpha^*$ is the minimum of the set $\{\alpha(\theta) : \theta\in[0,2\pi)\}$ and

$\beta^*$ is the minimum of the set $\{\beta(\theta) : \theta\in[0,2\pi)\}$,

then the value of $\alpha^* + \beta^*$ is

Let

$$P_1 = I = \begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix},\quad P_2 = \begin{bmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{bmatrix},\quad P_3 = \begin{bmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1\end{bmatrix},$$

$$P_4 = \begin{bmatrix}0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix},\quad P_5 = \begin{bmatrix}0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\end{bmatrix},\quad P_6 = \begin{bmatrix}0 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0\end{bmatrix}$$

and

$$X = \sum_{k=1}^6P_k\begin{bmatrix}2 & 1 & 3\\ 1 & 0 & 2\\ 3 & 2 & 1\end{bmatrix}P_k^T$$

where $P_k^T$ denotes the transpose of the matrix $P_k$. Then which of the following options is/are correct?

Let $x\in\mathbb R$ and let

$$P = \begin{bmatrix}1 & 1 & 1\\ 0 & 2 & 2\\ 0 & 0 & 3\end{bmatrix},\quad Q = \begin{bmatrix}2 & x & x\\ 0 & 4 & 0\\ x & x & 6\end{bmatrix}\quad\text{and}\quad R = PQP^{-1}.$$

Then which of the following options is/are correct?

Which of the following is(are) NOT the square of a $3 \times 3$ matrix with real entries?

How many $3 \times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there, for which the sum of the diagonal entries of $M^TM$ is 5?

If $A=\begin{bmatrix}2&-3\\-4&1\end{bmatrix}$, then adj $(3A^2+12A)$ is equal to :